Fuzzy inference system enabled neural network feedforward compensation for position leap control of DC servo motor

To improve dynamic performance and steady-state accuracy of position leap control of the direct current (DC) servo motor, a fuzzy inference system (FIS) enabled artificial neural network (ANN) feedforward compensation control method is proposed in this study. In the method, a proportional-integral-derivative (PID) controller is used to generate the baseline control law. Then, an ANN identifier is constructed to online learn the reverse model of the DC servo motor system. Meanwhile, the learned parameters are passed in real-time to an ANN compensator to provide feedforward compensation control law accurately. Next, according to system tracking error and network modeling error, an FIS decider consisting of an FI basic module and an FI finetuning module is developed to adjust the compensation quantity and prevent uncertain disturbance from undertrained ANN adaptively. Finally, the feasibility and efficiency of the proposed method are verified by the tracking experiments of step and square signals on the DC servo motor testbed. Experimental results show that the proposed FIS-enabled ANN feedforward compensation control method achieves lower overshoot, faster adjustment, and higher precision than other comparative control methods.

comprises four modules, that is, a PID controller, an ANN identifier, an ANN compensator, and an FIS decider that consists of an FI basic module and an FI finetuning module.
The PID controller with a clear principle and a simple algorithm is used as a baseline controller.Its main function is to maintain the control stability at the early control phase and produce learning samples for the ANN identifier.Furthermore, the ANN identifier is designed to online identify an inverse model of the DC servo motor system.Sharing the same network structure and learned parameters as the ANN identifier, the ANN compensator is constructed to provide accurate feedforward compensation for position control of the DC servo motor.
In particular, at the early control phase or the leap instantaneous of the tracking signal, the online samples are insufficient to train an effective ANN identifier.It can't learn the inverse model of the DC servo motor system accurately, thereby resulting in control uncertainty of the ANN compensator.Therefore, the FIS decider including an FI basic module and an FI finetuning module is developed to adaptively adjust the controlled quantity of the ANN compensator.The FI basic module and the FI finetuning module are applied to the coarse tuning and the finetuning of compensation quantity, respectively.Ultimately, after the combined effect of these modules, the proposed FIS-enabled feedforward compensation method can improve control performance as well as reduce control uncertainty.

Integrated control law
As shown in Fig. 1, the integrated control law of the proposed FIS-enabled ANN feedforward compensation method consists of the baseline control quantity u o and the compensated control quantity u c .Therefore, the overall control law u t can be expressed as follows The baseline control quantity u o is from the PID controller.Its expression is as follows where e θ (k) is the control error at k moment, e θ (k − 1) is the control error at k − 1 moment, t is the control sampling interval.Besides, k p , k i , and k d are the proportion, integration, and differentiation gains of the PID controller, respectively.
Additionally, as shown in Fig. 1, the compensated control quantity u c is equal to the reasoning output c of the FIS decider multiplied by the control output u n of the ANN compensator.Its expression is as follows The control output u n is calculated by the forward propagation of the ANN compensator.As illustrated in Fig. 2, it consists of one input layer with one neuron, two hidden layers with five neurons, and one output layer with one neuron.Its calculation process is as follows (1) where O 1 is the output of the input layer, O 2 i is the i-th output of the first hidden layer, O 3 j is the j-th output of the second hidden layer, and O 4 is the output of the output layer.In addition, σ (•) represents the Sigmoid activation function, ω 1 i is a weight between the input layer and the i-th neuron of the first hidden layer, b 2 i is a bias of the i-th neuron of the first hidden layer, ω 2 i,j is a weight between the i-th neuron of the first hidden layer and the j-th neuron of the second hidden layer, b 3 j is a bias of the j-th neuron of the second hidden layer, ω 3 i is a weight between the j-th neuron of the second hidden layer and the output layer, b 4 is a bias of the output layer.
In practice, the ANN compensator only performs forward propagation, and weights and biases are passed in real-time from the ANN identifier with the same network structure.These parameters are learned online by the backpropagation of the ANN identifier.To promote learning a precise inverse model of the controlled system, the Euclidean distance between the actual control law u t and the predicted control law u t is defined as the training loss function.Its expression is as follows In the backpropagation process, the weight ω and bias b of the ANN identifier are updated as follows where η is the learning rate, and γ is the momentum factor that is introduced to speed up the network conver- gence and prevent the network from falling into a local optimum during the updating process.
After the ANN identifier learns the reverse model of the DC servo motor online using the above backpropagation process, the learned parameters are passed in real-time to the ANN compensator to generate the initial compensation control law u n .Furthermore, to address the compensated uncertainty caused by the undertrained ANN, this study designs the FIS decider to adaptively generate the adjustment coefficient c for the ANN compensator.Its detailed principle is elaborated in the following section.
Finally, the algorithm of the FIS-enabled neural network feedforward compensation method is summarized in Table 1.

FIS decider design
The designed FIS decider consists of the FI basic module and the FI finetuning module, and its structure is illustrated in Fig. 3.The FI basic module is firstly designed to weaken the uncertainty impact caused by the undertrained ANN to the control system.It deduces the coarse adjustment coefficient c α for the compensated output of the ANN compensator according to the control error e θ and error change ec θ of the control system.Since the scarcity of training samples at the leap moment of the tracking signal and the random initialization of the network parameters, the weight change of the ANN identifier causes the control system to destabilize to a certain extent.Correspondingly, the FI finetuning module is designed to further generate the adjustment coefficient c β according to the control error e θ of the control system and the weight change dw θ of the ANN identifier.Finally, under the joint action of the two modules, the control uncertainty of the ANN compensator is adjusted.9) and ( 10)

FI basic module
The control error e and its change ec directly reflect the control performance of the DC servo motor system.
To facilitate the stability of the control system, they are selected as the input variables of the FI basic module.Compared to the conventional Mamdani fuzzy model, a T-S-type FI with a simple structure has the advantage of clear reasoning and fast operation 40 .Therefore, the T-S-type FI is introduced into the designed FIS decider to suppress the control uncertainty of the ANN compensator in this study.
To balance the rule knowledge compatibility and the inference sensitivity, the input variables e and ec are divided into seven fuzzy sets: {Negative Big ( NB ), Negative Medium ( NM ), Negative Small ( NS ), Zero ( ZE ), Positive Small ( PS ), Positive Medium ( PM ), Positive Big ( PB)}.The output variable is divided into five fuzzy sets: {Zero ( ZE ), Medium Small ( MS ), Medium ( M ), Medium Big ( MB ), Big Big ( BB)}.To accelerate inference calculation, the output variable adopts 0-order T-S-type FI and is represented by five constant values.Concretely, the universe of input variables e and ec are [ − 3, 3].The universe of the fuzzy output variable C α is [0,1], and the values corresponding to the five output fuzzy sets are shown in Table 2.
The triangular membership function is frequently employed in the FIS due to its simple structure and ease of calculation 41 .Therefore, to enhance the FI operation speed, all seven fuzzy sets of the input variables e and ec adopt triangular functions as membership functions, as shown in Fig. 4.
For the FI basic module, the rule base I is designed based on the following principles: At the initial moment of the control period and the leap moment of the tracking signal, the ANN identifier lacks sufficient training samples and can't learn the reverse model of the DC servo motor accurately.It leads the ANN compensator to hardly provide an efficient compensation quantity.The control error e and its change ec are large, resulting in a large uncertainty to the control system.Therefore, at this moment, the FI basic module is required to severely weaken the compensation output of the ANN compensator.
With the increase of online training samples, the ANN identifier learns the reverse model of the DC servo motor more accurately.The ANN compensator can provide more and more precise compensation quantity for Table 2.The fuzzy set division of the output variable of the FI basic module.www.nature.com/scientificreports/ the control system.At this stage, the control error e and its change ec gradually dwindle, thus the FI basic module needs to enhance the effect of the ANN compensator until the stability of the control system.According to the above reasoning principles, the fuzzy rule base of the FI basic module is given in detail in Table 3.
As the T-S-type FIS, the FI basic module uses the weighted sum method for defuzzification of the output variable.When the input variables e and ec activate R rules, the defuzzification of the output variable c α is expressed as follows where i is the weight of the i-th rule.Besides, E i (e) and EC i (ec) are the membership degrees of the input quanti- ties e and ec belonging to the fuzzy sets E i and EC i .
Since control error e and its change ec reflect the approximating level of the inverse model by the ANN iden- tifier passively, the FI basic module mainly adjusts the effect of the ANN compensator on the control system roughly.

FI finetuning module
To further adjust the compensation quantity of the ANN compensator as precisely as possible, the FI finetuning module is designed in the FIS decider.Considering the weight change of the ANN directly impacts the output of the learned inverse model, therefore, the change of the weights between the last hidden layer and the output layer is introduced in the FI finetuning module.In addition, control error is used as a supplement input.Specifically, the FI finetuning module takes the absolute value of control error and weight change as input variables.
Similar to the FI basic module, to balance the rule knowledge compatibility and the inference sensitivity, the fuzzy input variables |e| and |dω| are also divided into seven fuzzy sets: {Small Small ( SS ), Small Medium ( SM ), Small Big ( SB ), Medium ( M ), Big Small ( BS ), Big Medium ( BM ), Big Big ( BB)}.To adjust the compensation quantity of the ANN compensator more precisely, the output variable is divided into seven fuzzy sets: {Small Small ( SS ), Small Medium ( SM ), Small Big ( SB ), Medium ( M ), Big Small ( BS ), Big Medium ( BM ), Big Big ( BB )}.To accelerate inference calculation, the output variable still adopts 0-order T-S-type FI and is represented by seven constant values.Concretely, the universe of input variables |e| and |dω| are [0, 3].The universe of the fuzzy output variable C β is [0,1], the values corresponding to the seven output fuzzy sets are shown in Table 4.
Similarly, to enhance the FI operation speed, all seven fuzzy sets of the input variables |e| and |dω| still adopt triangular functions as membership functions, as shown in Fig. 5.
For the FI finetuning module, the rule base II is designed based on the following principles: On the one hand, when the state of |e| is the same as the state of |dω| , that is, when |e| is big, |dω| is also big, or when |e| is small, |dω| is also small, the state change of the control system is synchronized with the state change of the ANN identifier.The FI basic module can efficiently adjust the compensation output, no additional adjustments are required from the FI finetuning module.At this stage, the reasoning output of the FI finetuning module is close to 1.
On the other hand, when the state of |e| and the state of |dω| are inconsistent, first, when |e| is relatively big, but |dω| is relatively small, the training of the ANN is getting better and will converge soon, only minor (11)    suppression is required.On the contrary, when |e| is small, but |dω| is big, the ANN is poorly trained, resulting in uncertainty in the output of the ANN compensator.Thus, it is necessary to further suppress the compensation output appropriately.By leveraging the above reasoning principles, a total of forty-nine fuzzy rules can be obtained based on the input variables |e| and |dω| .Concretely, the fuzzy rule base II of the FI finetuning module is described in Table 5.
Similar to the FI basic module, the weighted sum method is applied for defuzzification of the output variable in the FI basic module uses.When the input variables |e| and |dω| activate R rules, the defuzzification of the output variable c β is expressed as follows (13)  where ϕ i is the weight of the i-th rule.|E| i (|e|) and DW i (|dω|) are the membership degrees of the input quantities |e| and |dω| belonging to the fuzzy sets |E| i and DW i .
Particularly, the reasoning output c β can essentially be considered an adaptive gain for the reasoning output c α of the FI basic module.Therefore, after inferencing the adjustment factors c α and c β , the initial compensation coefficient c o is obtained as follows Furthermore, to prevent violent jittering of the inference output, the initial compensation coefficient c o is followed by the saturation operation.Its calculation expression is as follows where ξ represents the saturation coefficient of the change ratio of the adjustment factor c(k).
After the saturation operation, the final adjustment factor c(k) is used to adaptively adjust the compensation quantity of the ANN-based feedforward compensator.It is applied to improve the compensation precision in the position leap control of the DC servo motor and guarantee the transient stability of the control system.

Experimental verification Experimental platform
To validate the effectiveness of the FIS-enabled ANN feedforward compensation method in real-time control, experimental research is conducted on the hardware-in-loop platform as shown in Fig. 6.It consists of a dual DC servo motor testbed, an integrated control box, and a computer.The motor testbed comprised an active motor and a driven motor.The control box includes a motor driver and a motor encoder.The motion control card is embedded in the computer through a PCI slot.In addition, the tracking controls of the step signal and square signal are carried out using the MATLAB/SIMULINK real-time workshop on the computer.Particularly, the DC servo motor used in the testbed is the T54H019 motor produced by Mingyago company, and its main parameters are shown in Table 6.
In the control experiment, the motion control card controlled by MATLAB/SIMULINK real-time workshop sends an analog voltage to the motor driver within the control box.The driver converts the analog voltage into the armature current of the motor, generating torque and controlling the rotation of the motor.
Additionally, in the process of motor movement, the encoder collects the real-time position of the motor and feeds it back to the motor driver.After signal processing, the motor driver transmits the motor position in the form of pulses to the motion control card.The computer displays the motor position in real-time through the MATLAB/SIMULINK real-time workshop.According to the difference between the actual position and the expected position, the proposed method run on the MATLAB/SIMULINK real-time workshop calculates the corresponding control quantity and performs real-time control of the DC servo motor. (15)

Position tracking results
Step signal tracking performance In this experiment, the PID parameters of the DC servo motor testbed are first adjusted to the optimal value as best as possible by using the trial-and-error method.Concretely, the PID parameters ( k p , k i , and k d ) of the motor position loop are set to 90, 0, and 20, and the PID parameters ( k p , k i , and k d ) of the motor speed loop are set to 8, 3, and 1.5, respectively.In particular, the PID parameters of the motor current loop can't be turned by MTALAB, thus they are selected as the default value inside the motor driver.Note that all the control methods adopt the same PID parameters in the step signal tracking experiments to make the comparison fairer.
In addition, the learning rate η and the momentum factor γ of ANN are set to η = 0.011 and γ = 0.20 , respectively.In the FI basic module, the quantization factors of fuzzy variables E and EC are set to 5.0 × 10 −3 and 5.0 × 10 −5 , respectively.In the FI finetuning module, the quantization factors of fuzzy variables E and DW are set to 5.0 × 10 −3 and 1.0 × 10 −5 , respectively.The saturation coefficient ξ of the change ratio of the adjustment factor is set to 0.005.Correspondingly, step signal tracking results using different control methods are depicted in Fig. 7, where ANN-PID represents the method consisting of the PID controller, the ANN identifier, and the ANN compensator.Besides, FIB-ANN-PID represents the method that adds the FI basic module based on ANN-PID.FIS-ANN-PID represents the proposed control method, that is, the method adds the FI finetuning module based on FIB-ANN-PID.
It can be seen from Fig. 7 that, in the step signal position tracking of the DC servo motor, the ANN-PID method lowers the overshoot and the setting time compared with the traditional PID method.Furthermore, the FIB-ANN-PID method and the FIS-ANN-PID method can improve the transient performance of the motor position control system compared to the ANN-PID method.Particularly, after introducing the FI finetuning module, the FIS-ANN-PID method realizes the small overshoot and the short settling time.Concretely, the quantitative comparison results of tracking step signals under different methods are summarized in Table 7, where SSME represents the steady-state mean error.As shown in Table 7, the ANN-PID method lowers SSME by 79.13%, 74.90%, and 78.69% compared to the PID method at the leap positions I, II, and III, respectively.It indicates that the ANN compensator can provide high-precision compensation for the position control of the DC servo motor when the control system into a steady state, benefitting from the ANN identifier can accurately learn the inverse model of the motor testbed online.When the ANN identifier can't be fully trained by the insufficient samples at the position leap moment, the output of the ANN compensator has a large uncertainty, thereby the ANN-PID method still has a big overshot.After introducing the single FI basic module, the FIB-ANN-PID method lowers the overshoot by 96.35%, 92.30%, and 90.86% compared with the PID method at the leap positions I, II, and III, respectively.The results show that the FI module can suppress the compensation uncertainty of the ANN compensator caused by the undertrained ANN identifier.
Furthermore, after adding the FI finetuning module, the FIS-ANN-PID method lowers the overshoot by 100% with little sacrifice in the settling time compared to the FIB-ANN-PID method at all leap positions.As shown in Figs. 8 and 9, the adjustment coefficient obtained by the combined action of the FI basic and finetuning module is more refined than that obtained by the single FI basic module at the position leap moment.It is because the FIS consisting of the FI basic and finetuning module considers not only the internal error and its change in the control system but also the change in the learned weights of the ANN identifier.
Consequently, the FIS-enabled ANN feedforward compensation method (i.e., FIS-ANN-PID) can not only enhance the dynamic quality but also improve the steady-state performance in the step signal tracking experiment of the DC servo motor.www.nature.com/scientificreports/

Square signal tracking performance
In the square signal tracking experiment, the PID parameters of the DC servo motor testbed are also adjusted to the optimal value as best as possible by using the trial-and-error method.Concretely, the PID parameters ( k p , k i , and k d ) of the motor position loop are set to 90, 0, and 20, and the PID parameters ( k p , k i , and k d ) of the motor speed loop are set to 12, 6, and 1.5, respectively.Similarly, the PID parameters of the motor current loop are also selected as the default value inside the motor driver considering they can't be turned by MTALAB/SIMULINK.Note that all the control methods adopt the same PID parameters in the square signal tracking experiments to make the comparison fairer.Additionally, the learning rate η and the momentum factor γ of ANN are set to η = 0.011 and γ = 0.12 , respectively.In the FI basic module, the quantization factors of fuzzy variables E and EC are set to 4.3 × 10 −3 and 6.7 × 10 −6 , respectively.In the FI finetuning module, the quantization factors of fuzzy variables E and DW are set to 4.3 × 10 −3 and 1.0 × 10 −5 , respectively.The saturation coefficient ξ of the change ratio of the adjust- ment factor is set to 0.005.Correspondingly, square signal tracking results using different control methods are drawn in Fig. 10.
It can be found from Fig. 10 that, compared to the traditional PID method, the ANN-PID method lowers the overshoot and the setting time in the square signal position tracking of the DC servo motor.By adding the FI module based on the ANN-PID method, both the FIB-ANN-PID method and the FIS-ANN-PID method can achieve a small overshoot and a short settling time of the motor position control system.Moreover, after introducing the FI finetuning module, the FIS-ANN-PID method can improve the dynamic performance of motor position control compared to the FIB-ANN-PID method.Quantitatively, the comparison results of tracking square signals under different control methods are summarized in Table 8.
As shown in Table 8, compared with the conventional PID method, the ANN-PID method lowers the overshoot by 38.94%, 40.78%, and 36.25%,reduces the settling time by 82.63%, 82.54%, 81.95%, and lowers the SSME by 54.37%, 55.13%, and 55.09% at positions I, II, and III, respectively.It testifies that the ANN compensator can implement a certain high-precision position compensation of the motor because the ANN identifier learns the inverse model of the motor testbed accurately online.But when the tracking signal suddenly changes, the training samples are significantly reduced, and the ANN identifier can't learn the inverse model of the motor testbed accurately.At this moment, the compensation output of the ANN compensator has a large uncertainty, resulting in the ANN-PID method still having a big overshot.
After introducing the FI basic module, the FIB-ANN-PID method lowers the overshoot by 88.33%, 85.66%, and 86.58%, reduces the settling time by 89.76%, 89.38%, and 89.53%, and lowers the SSME by 53.64%, 55.52%, and 56.96% compared to the PID method at positions I, II, and III, respectively.It verifies that when the ANN identifier can't learn the inverse model of the motor testbed effectively, the FI basic module can adjust the output uncertainty of the ANN compensator, thereby improving the dynamic performance of the motor control system.
However, the FI basic module inferences the adjustment coefficient according to the error and its change in the whole control system, it only considers the internal state of the control system rather than the internal state of the ANN, especially the change in the learning weights.Therefore, as displayed in Figs.11 and 12, compared with the single FI basic module designed in the FIB-ANN-PID method, the FIS decider consisting of the FI basic and finetuning modules can deduce the more imperceptible details.

Hyperparameter sensitivity
As shown in Eqs. ( 9) and ( 10), the learning rate η and the momentum factor γ are key parameters to make the ANN identifier learn the reverse model of the DC servo motor accurately.Take the square signal tracking experiment as an example, the sensitivity results of different learning rates and different momentum factors are shown in Fig. 13.It can be found from Fig. 13 that, when the learning rate is less than 0.011 in the square signal tracking experiment, the lower the learning rate, the larger the overshoot, and then the transient performance of the control system becomes poorer.When the learning is larger than 0.011, the overshoot increases with the increase in the learning rate.Therefore, the learning rate is determined as 0.011 in the tracking experiment.In addition, when the momentum factor is equal to 0.12, the overshoot of the control system is lowest, therefore, it is determined as the appropriate value in the tracking experiment.

Anti-interference performance
To evaluate the anti-interference performance of the proposed FIS-enabled ANN feedforward compensation control method, the disturbance experiments under tracking sawtooth and stochastic signals are carried out in the DC servo motor testbed.The experiment results are drawn in Fig. 14, where the two left subgraphs display the overall tracking results, and the two right subgraphs exhibit partial enlarged details.
As shown in Fig. 14a,b, the DC servo motor can quickly respond to the disturbing reference when tracking the sawtooth signal.Additionally, as shown in Fig. 14c,d  experiment of the stochastic signal, the DC servo motor also promptly adjusts to the altered reference.Therefore, the proposed control method demonstrates excellent anti-interference performance benefitting from the quick inference of the FIS decider as well as accurate compensation of the ANN compensator in real-time.

Conclusion
In this study, the FIS-enabled ANN feedforward compensation method is proposed to realize the high-performance position leap control of the DC servo motor.In the method, the ANN identifier is built to accurately learn the reverse model of the DC servo motor control system while the ANN compensator sharing the same network structure is designed to online provide high-precision feedforward compensation quantity.Furthermore, considering both the system tracking error and network modeling error, the FIS consisting of an FI basic module and an FI finetuning module is developed to adjust the ANN feedforward compensation quantity and prevent the uncertain disturbance of the undertrained ANN automatically.Experimental results show that when tracking the step and square signals, the proposed method lowers the overshoot by an average of 98.50% and 100%, reduces the settling time by an average of 8.85% and 18.89%, and lowers the steady-state error by an average of 33.51% and 19.17%, compared with the ANN method under the same conditions, respectively.It demonstrates the proposed method not only enhances the dynamic performance but also improves the steady-state accuracy of the DC servo motor control system significantly.Therefore, the FIS-enabled ANN feedforward compensation method is feasible and effective.Particularly, it is suitable for timevarying control systems that cannot be accurately modeled and easy to perform tracking control of the signals with abrupt change characteristics.
For future works, on the one hand, to guarantee the high synchronization performance of two DC servo motors 42 , a fuzzy inference-based ANN feedforward compensation method via integrating adaptive backstepping control 43 is worth being developed for dual DC servo motor systems.On the other hand, a variable universe fuzzy inference decided ANN feedforward compensation method is also worth further being investigated for the high-precision position control of DC servo motor systems.

Fig. 1 .
Fig. 1.The overall structure of the FIS-enabled ANN feedforward compensation method.

Fig. 3 .
Fig. 3.The overall structure of the designed FIS decider.

Fig. 4 .
Fig. 4. Membership functions in the FI basic module.(a) Membership function of fuzzy set E. (b) Membership function of fuzzy set EC.

Fig. 5 .
Fig. 5. Membership functions in the FI finetuning module.(a) Membership function of fuzzy set |E| .(b) Membership function of fuzzy set DW .

Fig. 6 .
Fig. 6.Experimental platform of the dual DC servo motor testbed.

Fig. 7 .
Fig. 7. Experimental tracking results of step signal under different control methods.(a) Overall tracking result.(b) Partial enlarged detail I. (c) Partial enlarged detail II.(d) Partial enlarged detail III.

Fig. 8 .
Fig. 8.The change curve of the adjustment coefficient in the step signal tracking experiment.(a) Overall reasoning result.(b) Partial enlarged detail.

Fig. 9 .
Fig. 9.The change curve of the finetuning coefficient in the step signal tracking experiment.(a) Overall reasoning result.(b) Partial enlarged detail.

Fig. 10 .
Fig. 10.Experimental tracking results of square signals under different control methods.(a) Overall tracking result.(b) Partial enlarged detail I. (c) Partial enlarged detail II.(d) Partial enlarged detail III.

Fig. 11 .Fig. 12 .
Fig. 11.The change curve of the adjustment coefficient in the square signal tracking experiment.(a) Overall reasoning result.(b) Partial enlarged detail.

Fig. 13 .
Fig. 13.Sensitivity results of different learning rates and different momentum factors.(a) Sensitivity of different learning rates.(b) Sensitivity of different momentum factors.

Fig. 14 .
Fig. 14.Anti-interference experiment results of under sawtooth and stochastic signals.(a) Overall tracking result under sawtooth signal.(b) Partial enlarged detail under sawtooth signal.(c) Overall tracking result under stochastic signal.(d) Partial enlarged detail under stochastic signal.

Table 1 .
The algorithm of the proposed control method.

Table 3 .
The fuzzy rule base I of the FI basic module.

Table 4 .
The fuzzy set division of the output variable of the FI finetuning module.

Table 5 .
The fuzzy rule base II of the FI finetuning module.

Table 6 .
Main parameters of the DC servo motor used in the testbed.

Table 7 .
Experimental comparison results of tracking step signals.Significant values are given in bold.

Table 8 .
Experimental comparison results of tracking square signal.Significant values are given in bold.